A survey of some recent results on truncated Toeplitz operators

Abstract

Truncated Toeplitz operators are compressions of Toeplitz operators on model spaces; they have received much attention in the last years. This survey article presents several recent results, which relate boundedness, compactness, and spectra of these operators to properties of their symbols. We also connect these facts with properties of the natural embedding measures associated to these operators.2010 Mathematics Subject Classification. 30J05, 30H10, 46E22.

Point spectrum and hypercyclicity problem for a class of truncated Toeplitz operators

“…Assume that λ is an eigenvector and so there exists a nontrivial f ∈ K θ satisfying (5). If |z j | < 1, then the right-hand side of (5) must vanish at z j and we get z N j θ(z j )h(z j ) + R(z j ) = 0.…”

“…During the last decade the theory of truncated Toeplitz operators became an active field of research (see the surveys [5,7] and the references therein). However, it seems that the spectral properties of TTO-s are far from being well understood.…”

Symbols of compact truncated Toeplitz operators

“…Motivated by these findings, a similar study in to the symbols of compact truncated Toeplitz operators was initiated. Section 5 of [6] gives an overview of many results in this area. In particular, the role played by bounded symbols in the case of bounded truncated Toeplitz operators seems to be replaced by symbols of the form IC(T) when we are considering compact truncated Toeplitz operators.…”

Laplace–Carleson Embeddings on Model Spaces and Boundedness of Truncated Hankel and Toeplitz Operators

“…In general, results for other Schatten norms (even the Hilbert-Schmidt norm) are harder to come by, as mentioned in [4]. The interested reader may also consult [13].…”